nteg rator with Time delay Feed fo A(q)=q(q-1) Easy to inclu de feedforward B(a)=h-r)+T=h-q+ Estimate paramete rs Cq=d+g yt+ d =R (q)u,(t+S(q()(t+Si,qr),(t Minimum phase if T <h/2. Cont roller wit h d= 1, T changed from 0. 4 to 0 vf filtered fee dfo rward signal (a)si,y Cont rol law R (q()u(t=5(()yt-Si,(q()v(t Fee dfo rward has proven ve ry useful in applica tion Cont roller with d= 2 Dis )计M Command signals can also be inc hu ded R (q( )u(t =T(q()u(t-5(()yt e command signal (set point, re fe rence signal) Command signals and fee dfo rward can be combined observat i Indirect self-tuners re quire estimation of Direct self-tuners have unexpectedly nice pro pe rtles . Self-tuners drive cova rances to zero Com pare PI cont rol The number of covariances de pend on the paramete rs Wit h sufficiently many paramete rs we btain The parameters do not necessarily c Design paramete rs are pre dictions horizon d, sampling pe riod ane Rand s polynomia ls It is easy to inclu de fee forward .Easy to c hec h in o pe ration . Pe rformance assessment O K. Ast rom and B WittenmarkIntegrator with Time Delay A(q) = q(q ￾ 1) B(q)=(h ￾  )q +  = (h ￾  )(q +  h ￾  ) C(q) = q(q + c) Minimum phase if  < h=2. Controller with d = 1,  changed from 0.4 to 0.6 at time 100. 0 100 200 300 400 −5 0 5 0 100 200 300 400 −20 0 20 Time Time (a) y u Controller with d = 2 0 100 200 300 400 −5 0 5 0 100 200 300 400 −20 0 20 Time Time (b) y u Feedforward Easy to include feedforward! Estimate parameters in y(t + d) = R (q￾1 )uf (t) + S (q￾1 )yf (t) + S f f (q￾1 )vf (t) vf ltered feedforward signal Control law R^ (q￾1 )u(t) = ￾S^ (q￾1 )y(t) ￾ S^ f f (q￾1 )v(t) Feedforward has proven very useful in applica￾tions! Discuss why! Command signals can also be included R^ (q￾1 )u(t) = T (q￾1 )uc(t) ￾ S^ (q￾1 )y(t) uc command signal (set point, reference signal) Command signals and feedforward can be combined Observations  Indirect self-tuners require estimation of C-polynomial  Direct self-tuners have unexpectedly nice properties  Self-tuners drive covariances to zero  Compare PI control  The number of covariances depend on the parameters  With suciently many parameters we obtain moving average control  The parameters do not necessarily con￾verge  Design parameters are predictions horizon d, sampling period and number of parame￾ters in R and S polynomials  It is easy to include feedforward  Easy to check in operation  Performance assessment c K. J. Åström and B. Wittenmark 6